Greedy Strikes Back: Improved Facility Location Algorithms

Guha, Sudipto; Khuller, Samir

doi:10.1006/jagm.1998.0993

Cited by 479 publications

(399 citation statements)

References 18 publications

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“…We show that if the instance is (α, ǫ)-perturbation resilient with α > 2 + √ 3, then we can in polynomial time output a clustering that provides a (1 + 5ǫ/ρ)-approximation to the optimum, where ρ is the fraction of the points in the smallest cluster. Thus this improves over the best worst-case approximation guarantees known [20] when ǫ ≤ √ 3ρ/5 and also beats the lower bound of (1 + 1/e) on the best approximation achievable on worst case instances for the metric k-median objective [17,18] when ǫ ≤ ρ/(5e).…”

Section: (α ǫ)-Perturbationsupporting

confidence: 58%

Clustering under Perturbation Resilience

Balcan

Liang

2012

Automata, Languages, and Programming

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Abstract. Motivated by the fact that distances between data points in many real-world clustering instances are often based on heuristic measures, Bilu and Linial [13] proposed analyzing objective based clustering problems under the assumption that the optimum clustering to the objective is preserved under small multiplicative perturbations to distances between points. The hope is that by exploiting the structure in such instances, one can overcome worst case hardness results.In this paper, we provide several results within this framework. For center-based objectives, we present an algorithm that can optimally cluster instances resilient to perturbations of factor (1 + √ 2), solving an open problem of Awasthi et al. [3]. For k-median, a center-based objective of special interest, we additionally give algorithms for a more relaxed assumption in which we allow the optimal solution to change in a small ǫ fraction of the points after perturbation. We give the first bounds known for k-median under this more realistic and more general assumption. We also provide positive results for min-sum clustering which is typically a harder objective than center-based objectives from approximability standpoint. Our algorithms are based on new linkage criteria that may be of independent interest.Additionally, we give sublinear-time algorithms, showing algorithms that can return an implicit clustering from only access to a small random sample.Key words. clustering, perturbation resilience, k-median clustering, min-sum clustering AMS subject classifications. 68Q25, 68Q32, 68T05, 68W25, 68W401. Introduction. Problems of clustering data from pairwise distance information are ubiquitous in science. A common approach for solving such problems is to view the data points as nodes in a weighted graph (with the weights based on the given pairwise information), and then to design algorithms to optimize various objective functions such as k-median or min-sum. For example, in the k-median clustering problem the goal is to partition the data into k clusters C i , giving each a center c i , in order to minimize the sum of the distances of all data points to the centers of their cluster. In the min-sum clustering approach the goal is to find k clusters C i that minimize the sum of all intra-cluster pairwise distances. Yet unfortunately, for most natural clustering objectives, finding the optimal solution to the objective function is NP-hard. As a consequence, there has been substantial work on approximation algorithms [18,14,9,15,1] with both upper and lower bounds on the approximability of these objective functions on worst case instances.Recently, Bilu and Linial [13] suggested an exciting, alternative approach aimed at understanding the complexity of clustering instances which arise in practice. Motivated by the fact that distances between data points in clustering instances are often based on a heuristic measure, they argue that interesting instances should be resilient to small perturbations in these distances. In particular, if small pertur...

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Section: (α ǫ)-Perturbationsupporting

confidence: 58%

Clustering under Perturbation Resilience

Balcan

Liang

2012

Automata, Languages, and Programming

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“…These two results are very close to the best possible approximation factor in that [GUHA98] have shown that we cannot get an approximation factor below 1.463 in polynomial time unless NP ⊆ DTIME[n O(log log n) ].…”

Section: Connection Costsupporting

confidence: 74%

A Clustering-Based Selective Probing Framework to Support Internet Quality of Service Routing

Jariyakul

Znati

2005

Distributed Computing – IWDC 2005

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The advent of the multimedia applications has triggered widespread interest in QoS supports. Two Internet-based QoS frameworks have been proposed: Integrated Services (IntServ) and Differentiated Services (DiffServ). IntServ supports service guarantees on a per-flow basis. The framework, however, is not scalable due to the fact that routers have to maintain a large amount of state information for each supported flow. DiffServ was proposed as an alternate solution to address the lack of scalability of the IntServ framework. DiffServ uses class-based service differentiation to achieve aggregate support for QoS requirements. This approach eliminates the need to maintain per-flow states on a hop-by-hop basis and reduces considerably the overhead routers incur in forwarding traffic.Both IntServ and DiffServ frameworks focus on packet scheduling. As such, they decouple routing from QoS provisioning. This typically results in inefficient routes, thereby limiting the ability of the network to support QoS requirements and to manage resources efficiently. The goal of this thesis is to address this shortcoming. We propose a scalable QoS routing framework to identify and select paths that are very likely to meet the QoS requirements of the underlying applications. The tenet of our approach is based on seamlessly integrating routing into the DiffServ framework to extend its ability to support QoS requirements. Scalability is achieved using selective probing and clustering to reduce signaling and routers overhead.The major contributions of this thesis are as follows: First, we propose a scalable routing architecture that supports QoS requirements. The architecture seamlessly integrates the QoS traffic requirements of the underlying applications into a DiffServ framework. Second, we propose a new delay-based clustering method, referred to as d-median. The proposed clustering method groups Internet nodes into clusters, whereby nodes in the same cluster exhibit equivalent delay characteristics. Each cluster is represented by anchor node. Anchors use selective probing to estimate QoS parameters and select appropriate paths for traffic forwarding.A thorough study to evaluate the performance of the proposed d-median clustering algorithm is conducted. The results of the study show that, for power-law graphs such as the Internet, the d-median clustering based approach outperforms the set covering method commonly proposed in the literature. The study shows that the widely used clustering methods, such as set covering or k-median, are inadequate to capture the balance between cluster sizes and the number of clusters. The results of the study also show that the proposed clustering method, applied to power-law graphs, is robust to changes in size and delay distribution of the network. Finally, the results suggest that the delay bound input parameter of the d-median scheme should be no less than 1 and no more than 4 times of the average delay per one hop of the network. This is mostly due to the weak hierarchy of the Internet resulting fr...

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“…The algorithmic concepts can be based on LP-rounding [15,16] or on primal-dual approach [17,18]. The 1.52 approximation ratio that has been obtained in most recent studies is close to the lower bound proved in [19]. Many variants of FLP have been studied; most of them are NP-hard [20].…”

Section: Related Workmentioning

confidence: 86%

Optimal network locality in distributed virtualized data-centers

Leblet

Simon

et al. 2011

Computer Communications

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Cost efficiency is a key aspect in deploying distributed service in networks within decentralized service delivery architectures. In this paper, we address this aspect from an optimization and algorithmic standpoint. The research deals with the placement of service components to network sites, where the performance metric is the cost for acquiring components between the sites. The resulting optimization problem, which we refer to as the k-Component Multisite Placement Problem, is applicable to service distribution in a wide range of communication networking scenarios. We provide a theoretical analysis of the problem's computational complexity, and develop an integer programming model for providing reference results for performance benchmarking. On the algorithmic side, we present four approaches: an algorithm with approximation guarantee and three heuristics algorithms. The first heuristic is derived from graph theory on domatic partition. The second heuristic, built on intuition, admits distributed computation. The third heuristic emphasizes on fairness in cost distribution among the sites. We report simulation results for sets of networks where cost is represented by round-trip time (RTT) originating from real measurements. For small networks, the integer model is used to study algorithm performance in terms of optimality. Large networks are used to compare the al- * Corresponding author1 The work of this author is supported by CENIIT, Linköping University, Sweden 2 The work of this author is supported by Vipeer project and Agence Nationale de la Recherche, France February 11, 2011 gorithms relatively to each other. Among the algorithms, the heuristic based on intuition has close-to-optimal performance, and the fairness heuristic achieves a good balance between single-site cost and the overall one. In addition, the experiments demonstrate the significance of optimization for cost reduction in comparison to a the random allocation strategy. Preprint submitted to Elsevier

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Greedy Strikes Back: Improved Facility Location Algorithms

Cited by 479 publications

References 18 publications

Clustering under Perturbation Resilience

Clustering under Perturbation Resilience

A Clustering-Based Selective Probing Framework to Support Internet Quality of Service Routing

Optimal network locality in distributed virtualized data-centers

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